An Inductive Approach to the Hodge Conjecture for Abelian Varieties

Let X be a smooth complex projective variety of dimension g. A Hodge class of degree 2d on X is, by definition, an element of H(X,Q)∩H(X). The cohomology class of an algebraic subvariety of codimension d of X is a Hodge class of degree 2d. The original Hodge conjecture states that any Hodge class on X is algebraic, i.e., a Q-linear combination of classes of algebraic subvarieties of X. Lefschetz’ Theorem says that Hodge classes of degree 2 are always algebraic. The classical Hodge conjecture has been generalized by Grothendieck as follows (see Steenbrink [S] page 166). To fix some notation, we will always designate a Hodge structure by its rational vector space V , the splitting V ⊗ C = ⊕p+q=mV p,q being implicit. We say that V is effective if V p,q = 0 when either p or q is negative. Recall that the level of a Hodge structure is the integer Max{|p− q| : V p,q 6= 0}